On the equivalence of the sum and the maximal term of the Dirichlet series absolutely convergent in the half-plane

Abstract

For absolutely convergent in the half-plane ${zcolon {mRe,}z<0}$ Dirichlet series $F(z)=sumlimits_{n=0}^{+infty}a_ne^{zlambda_n},$ where $0leqlambda_nuparrow +infty (0leq nuparrow+infty),$ we establish conditions on the coefficients of itsNewton majorant, sufficient for the relation $F(x+iy)=(1+o(1))a_{u(x)}e^{(x+iy)lambda_{u(x)}}$ to hold as$xo -0$ outside some set $E$ of zero logarithmic density in thepoint $0,$ uniformly by $yin{mathbb R}$.